On dense embeddings of discrete Abelian groups into locally compact groups (Q1420727)

In 1940 A. Weil established an interesting property of the group of integers: the image of every embedding of \(\mathbb Z\) into a locally compact group is either discrete or precompact. In the present paper, the following result is proved: An infinite Abelian group \(G\) satisfies that every embedding of \(G\) into a locally compact group is either discrete or precompact if and only if \(G\) is isomorphic to either \(\mathbb Z\times F\) or \(\mathbb Z_{p^{\infty}}\times F\), where \(F\) is a finite group and \(\mathbb Z_{p^{\infty}}\) is the group of roots of unity whose degrees are powers of a prime \(p\).